An infinite VC dimensional space vs using hierarchical subspaces of finite but growing VC dimensions
In the first case, the VC dimension of $H$ being infinite implies that $H$ is not (agnostic) PAC learnable (see p. 48 of Understanding Machine Learning: From Theory to Algorithms). So, in general, your classifier is not guaranteed to succeed.

In the second case, your division of $H$ implies that $H$ is nonuniformly learnable (chapter 7 of the cited book). This implies that you can get a generalization bound by using structural risk minimization.